Exercise 3.24

(a) Let $\equiv$ be the relation. Assume $p_n\equiv q_n$ and $q_n\equiv r_n$. We know that $d(p_n,p_n)\to 0$ whenever $p$ is Cauchy, so that $p_n\equiv p_n$. By the symmetry of $d$, it is clear that $q_n\equiv p_n$. By the triangle inequality,

which shows that $d(p_n,r_n)\to 0$ and $p_n\equiv r_n$. This shows that $\equiv$ is an equivalence relation.

(b) Let $p_n\equiv p_n^{\prime }$. It suffices to show that $\lim <!--\; FUNCTION\; APPLICATION\; -->d(p_n,q_n)=\lim <!--\; FUNCTION\; APPLICATION\; -->d(\underset{n}{\overset{\prime }{p}},q_n)$.

Consequently, since $d(p_n,p_n^{\prime })\to 0$, $\lim <!--\; FUNCTION\; APPLICATION\; -->d(p_n,q_n)\le \lim <!--\; FUNCTION\; APPLICATION\; -->d(\underset{n}{\overset{\prime }{p}},q_n)$. By interchanging $p_n$ and $p_n^{\prime }$, we also get that $\lim <!--\; FUNCTION\; APPLICATION\; -->d(\underset{n}{\overset{\prime }{p}},q_n)\le \lim <!--\; FUNCTION\; APPLICATION\; -->d(p_n,q_n)$, which proves equality. $\mathrm{\Delta }$ is therefore well-defined. We now need to show that it is a metric. It is clear that $\mathrm{\Delta }$ is positive, and that $\mathrm{\Delta }(P,Q)=0$ iff $P=Q$. The symmetry of $\mathrm{\Delta }$ follows from the symmetry of $d$. It remains to be shown that $\mathrm{\Delta }$ satisfies the triangle inequality.

(c) We first prove a lemma: Let $r_k$ be a sequence, and define the family of equivalence classes $Q_k=[q_k]$ by letting $q_k$ be constant sequences given by $q_{k,n}=r_k$ for all $n$. If $r_n$ is Cauchy, then $Q_k$ converges to the equivalence class $R=[r]$. To see this, choose an $𝜖>0$ and pick $N$ such that $d(r_k,r_{\ell })<𝜖$ whenever $k,\ell >N$. Then

Consequently

as desired.

Now for the main result, let $P_k=[p_k]$ be a Cauchy sequence in $X^{\text{∗}}$. For each $k$, choose a number $N(k)$ such that $d(p_{k,m},p_{k,n})<\frac{1}{2^k}$ whenever $m,n>N(k)$, and let $r_k=p_{k,N(k)}$. We now want to show that $r$ is Cauchy: Choose an $𝜖>0$. First remember that $P_i$ is Cauchy, so that $\mathrm{\Delta }(P_m,P_n)=\underset{k}{\lim <!--\; FUNCTION\; APPLICATION\; -->}d(p_{m,k},p_{n,k})<\frac{𝜖}{6}$ when the indices are sufficiently large. The convergence of that limit means that $d(p_{m,s},p_{n,s})<\frac{𝜖}{3}$ whenever $s$ is sufficiently large. Also note that $\frac{1}{2^m}<\frac{𝜖}{3}$ and $\frac{1}{2^m}<\frac{𝜖}{3}$ whenever $m$ and $n$ are sufficiently large. Choose $m,n$ sufficiently large to fulfill these criteria simultaneously. Then

So $r$ is Cauchy as desired.

This allows us to reach our desired conclusion: Define $Q_k$ in terms of $r$ as we did above. Now note that $\mathrm{\Delta }(P_k,Q_k)<\frac{1}{2^k}$, so that $\mathrm{\Delta }(P_k,Q_k)\to 0$. Since $Q$ converges to $R$, so does $P$. Consequently, any Cauchy sequence in $X^{\text{∗}}$ converges. This makes $X^{\text{∗}}$ complete.

(d)

(e) Let $R=[r]$ be any element of $X^{\text{∗}}$. We have shown that $Q_k$ as defined above is a sequence in $X$ that converges to $R$. This implies that $X$ is dense in $X^{\text{∗}}$.